Better idea of what you have helps!
A pretty big guess but I might lean toward it being a fitting that has frozen? But then I also don't know where you are and how cold it got?
Finding the tanks not drained might mean it doesn't get that cold? Or does it mean the other owner just did not do things right to keep it from freezing?
Doing this remote stuff leaves lots of questions, at times!
Does this seem to be somewhat right on where the black and grey tanks set?
Lots of guess involved in that drawing but some of my thinking may help with what I'm seeing?
I don't spot on this drawing any drain valve in that area but it may be there! I normally think of the drains as being inline with a stub of pipe sticking out underneath so that when you open the valve, the water passes all the way down below the cover underneath before dropping out.
Is there any way to look from underneath to spot a stub of pipe sticking out for the drain? Or maybe an access hole left in the floor under the vanity so you could pull it up and peek at the pipes?
My two thoughts are that the handle or valve if there may be broken so that when the handle moves, the inside doesn't close and water runs out the drain?
Or is it a broken pipe that didn't get drained and froze? Those pipes are often above the bottom cover but below the floor level inside. Maybe just running across underneath the vanity and you might find a flap to remove?
Getting a look at things is the hard part but some ideas may help.
I think of the pipe or tubing as pretty flexible and the fittings are often harder plastic that breaks quicker. When I've looked at frozen, it seems the joints at fitting are more likely to be the problem.
Just as an idea, can you get a look at the places where the vanity pipes connect to the main run going left/right under it?
Or are you in far South Florida and I'm nuts for thinking about frozen pipes?
Are you needing to get this winterized soon or just needing it fixed?
For quicker, using air to blow out the lines might get it done without having to fix a leaking pipe before next week?